Fragmentation of the photoabsorption strength in neutral and charged metal microclusters
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Fragmentation of the photoabsorption strength in neutral and charged metalmicroclusters
C. Yannouleas ∗ and R.A. Broglia Dipartimento di Fisica, Universita di Milano, and INFN, Sez. Milano,I-20133 Milano, Italy, and The Niels Bohr Institute, DK-2100 Copenhagen Ø, Denmark
M. Brack † The Niels Bohr Institute, DK-2100 Copenhagen Ø, Denmark
P.-F. Bortignon
Istituto di Ingegneria Nucleare, CESNEF Politecnico di Milano, Italy, and INFN, LNL, Legnaro, Italy (Dated: 16 December 1988; Phys. Rev. Lett. , 255 (1989))The line shape of the plasma resonance in both neutral and charged small sodium clusters iscalculated. The overall properties of the multipeak structure observed in the photoabsorption crosssection of spherical Na and Na neutral clusters can be understood in terms of Landau damping.Quantal configurations are shown to play an important role. In the case of charged Na +9 and Na +21 clusters a single peak is predicted that carries most of the oscillator strength. PACS numbers: 36.40.+d, 31.50.+w, 33.20.Kf
Recent photoabsorption experiments [1] have revealedthe surface plasmon in small neutral sodium clusters. Aqualitative overall account of the dependence of the res-onance frequencies on the cluster size can be achieved interms of the extended ellipsoidal shell model [2, 3] andof the experimental static polarizabilities. The reportedline broadening seems to obtain, as suggested in ref. [1],an important contribution from the coupling of the dipoleresonance to quadrupole shape fluctuations of the cluster[4, 5].Due to the fact that no complete photoabsorptioncurve has yet been experimentally determined, the de-tailed shapes and widths of the resonances are still ratheruncertain. In particular, the contribution of direct plas-mon decay mechanisms to the line widths must be inves-tigated, especially in the case of small clusters, in viewof the detailed microscopic calculations carried out byEkardt [6] and Beck [7] within the framework of the time-dependent local density approximation (TDLA).In the present paper, we study the structure ofthe plasmon resonance within the mean-field frameworkmaking use of the random phase approximation (RPA).Because these studies can most clearly be done in thecase of spherical clusters, we presently restrict our analy-sis to the neutral Na and Na and to the charged Na +9 and Na +21 . It will be concluded that, in the case of theneutral Na and Na , Landau damping is important andthat the observed line shapes are the result of a detailedinterplay between quantum size effects and residual in-teractions among particle-hole excitations, as well as ofthe thermal fluctuations of the cluster shapes. In the caseof the charged Na +9 and Na +21 , Landau damping is prac-tically absent and the dipole strength essentially consistsof a single peak exhausting most of the plasmon oscillatorstrength. The calculations to be discussed below parallel thosecarried out in studies of giant resonances in nuclei (cf.,i.e. [8]). A discrete particle-hole basis of dimension N isconstructed and the Hamiltonian H = H + V, (1)sum of the Hatree-Fock Hamiltonian H and the residualinteraction V , is diagonalized using the RPA. In what FIG. 1: Self-consistent potentials [10] associated with Na and Na +21 and used in the solution of the single-particle Hamil-tonian H appearing in eq. (1). For Na , the resulting single-particle levels relevant to the discussion of the dipole reso-nance, as well as some of the associated unperturbed particle-hole transitions, are also shown. follows, we shall specify the single-particle potentials self-consistently in the spherical jellium-background model [9]using the density variational formalism in a semiclassicalapproximation [10]. These potentials are displayed inFig. 1.The residual two-body interaction V is given by V ( | ~r − ~r | ) = e | ~r − ~r | + dV xc [ ρ ] dρ δ ( | ~r − ~r | ) . (2)Here V xc [ ρ ] = d E xc [ ρ ] /dρ is the exchange-correlation po- tential in the ground state. As in refs. [9, 10], we use theexchange-correlation energy density E xc [ ρ ] of Gunnarssonand Lundqvist [11].The RPA equations (see, e.g., ref. [12]) (cid:18) A BB ∗ A ∗ (cid:19) (cid:18) X n Y n (cid:19) = E n (cid:18) X n − Y n (cid:19) , are written in terms of the angular-momentum coupledmatrix elements of the interaction (2), i.e. A ( ph, p ′ h ′ ) − ( ǫ n p ,l p − ǫ n h ,l h ) δ l p ,l p ′ δ l h ,l h ′ δ n p ,n p ′ δ n h ,n h ′ = 2 R ( ph ′ , hp ′ ) ( − ) l p + l p ′ × [(2 l p + 1)(2 l h + 1)(2 l p ′ + 1)(2 l h ′ + 1)] (2 λ + 1) (cid:18) l h λ l p (cid:19) (cid:18) l h ′ λ l p ′ (cid:19) = ( − ) λ B ( ph, p ′ h ′ ) , (3)where R ( ph ′ , hp ′ ) = Z r dr r dr R n p ,l p ( r ) R n h ′ ,l h ′ ( r ) V ( r , r ; λ ) R n h ,l h ( r ) R n p ′ ,l p ′ ( r ) , (4)and where R n i ,l i ( r ) is the radial part of single-particlewave functions. The radial contribution of the two-bodyinteraction (2) in multipole order λ is given by V ( r , r ; λ ) = e rλ < rλ +1 > + dV xc [ ρ ] dρ δ ( r − r ) r λ + 14 π , where r < = min ( r , r ) and r > = max ( r , r ).The indices n i appearing in eqs. (3) and (4) denotethe number of nodes for the corresponding single-particlestates. The orbital angular momenta of the particles andholes participating in the excitations are denoted by l i ,the total angular momentum of the excitation being λ ,which in the present calculation is set equal to 1 (dipolevibration). The 3j symbols appearing in (3) take propercare of the angular momentum coupling, as well as of theparity conservation conditions. The factor 2 accounts forthe spin degeneracy. The RPA eigenvectors are written as a linear combi-nation of particle-hole excitations in terms of the for-wardsgoing and backwardsgoing amplitudes, X n ( ph ) and Y n ( ph ) respectively, according to | n i = X ph [ X n ( ph ) | ( ph − ) λ µ i− ( − ) λ + µ Y n ( ph ) | ( hp − ) λ , − µ i ] , (5)where a singlet spin configuration is implied.The dipole transition probabilities associated with thestate (5) can be written as B ( E , → n ) = 23 |h n ||M ( E || i| , where h n ||M ( E || i = X ph h X ∗ n ( ph ; 1) + ( − ) λ Y ∗ n ( ph ; 1) i h p ||M ( E || h i , (6)are reduced matrix elements [13] of the dipole operator M ( E µ ) = p π/ er Y µ (ˆ r ). The radial wave functions R n i ,l i are calculated by di-agonalizing the single-particle Hamiltonian H in a ba-sis including N = 25 harmonic oscillator major shells.(For the oscillator parameter of this basis, we have used¯ hω = 1 . eV . The results do not, however, depend onthis choice.)The associated unperturbed particle-hole excitations,examples of which are displayed in Fig. 1, exhaustthe Thomas-Reiche-Kuhn sum rule S ( E
1) = N ¯ h e / m ,that is X ph ( ǫ n p ,l p − ǫ n h ,l h ) | p / h p ||M ( E || h i| = S ( E , where ǫ n i ,l i are the single-particle energies. This resultis also valid for the correlated eigenstates (5) and associ-ated eigenvalues E n , since the RPA preserves the energy-weighted sum rule (EWSR).In Fig. 2 we display the oscillator strength functionsversus excitation energy for the RPA dipole in the caseof the neutral Na and Na clusters, as well as in thecase of the charged Na +9 and Na +21 clusters. The unper-turbed oscillator strength for the neutral clusters is alsodisplayed.A conspicuous feature of the RPA results for the neu-tral clusters is the sizeable amount of Landau damping.In fact, the variances of these response functions are σ ≈ . eV in both cases implying a ratio σ/ ¯ E ≈ . E here being the energy centroids of the resonances.Nonetheless, the identification of the collective states isquite unique. Indeed, in the case of Na , there is a singlestate at ≈ . eV exhausting ≈
75% of the EWSR, whileabout the same strength is distributed among two lineslocated at 2 . eV and 2 . eV in the case of Na . Asbefitted collective states, the associated wave functionsexhibit several (about ten) forwardsgoing amplitudes (X-components) that are larger than 0.1. Also, most of thebackwardsgoing amplitudes (Y-components) contributeconstructively to the transition amplitude (6). The wavefunctions associated with the rather weak states whichare strongly red shifted with respect to the collectivestates are, on the other hand, dominated by a coupleof components. We note that the results shown in themiddle of Fig. 2 for Na display an extent of fragmen-tation similar to the TDLDA calculation by Ekardt (cf.Fig.6 of ref. [6]).The plasmon strength function is drastically modifiedin the case of the charged Na +9 and Na +21 clusters, wherea single peak lying at ≈ eV (Na +9 ) and at ≈ eV (Na +21 ) exhausts ≈
93% and ≈
83% of the EWSR,respectively. This result seems to be consistent with re-cent experimental findings reported[14] for the potassiumclusters K +9 and K +21 .The difference between positively charged and neutralclusters is due to the difference in the associated aver-age potentials (cf. Fig. 1). Indeed, the potentials forthe charged clusters are deeper than the potentials forthe corresponding neutral clusters (observe that the po-tential for the neutral clusters exhibit an almost constant FIG. 2: Oscillator strength function for the photoabsorptionof Na , Na , Na +9 , and Na +21 clusters. In the upper figures,the unperturbed oscillator strengths for the neutral clustersis displayed. Also, in the upper figures, the strength valuesabove 3 eV have been multiplied by 100. central depth at ≈ − . eV ). For Na +9 the increase of thepotential depth at the center amounts to ≈ +21 the corresponding increase is ≈ E agreewithin 2 − α (givenby the negative energy-weighted sum rules) agree within4 − / ¯ hω ≈ . and Na , re-spectively. The results are shown in Fig. 3 in comparisonwith the experimental data [16]. Aside from a shift of ≈ FIG. 3: Photoabsorption cross sections per atom (for Na and Na ) resulting from the folding of the RPA oscillatorstrengths with Lorentzian shapes normalized to unity areshown (solid line). The widths of the Lorentzians are speci-fied by Γ / ¯ hω ≈ . and Na , respectively[5], ¯ hω being the peak energy of a given Lorentzian. Theexperimental data[16] are also shown. The dashed line is thesolid line shifted so that there is a maximum overlap with thedata. sections is well reproduced. In particular for Na , thedip seen in the experimental cross section near 460 nm ,as well as the weak bump in Na near 590 nm , seem tocome out of the present RPA description.The shift of the calculated dipole peaks with respect tothe observed values for the neutral clusters is consistentwith the fact that the predicted static dipole polarizabil-ities within the LDA model [9, 10, 17] are lower than theexperimental ones by ≈ /r Coulomb behavior of the meanfield. As a consequence, the tail of the electron densities (the so-called ’spill-out’) is underestimated; since the lat-ter is known to be correlated to both the dipole polariz-abilities [17] and the surface plasmons [10], these quan-tities are also underestimated. Indeed, a ’self-interactioncorrection’ aimed at a better treatment of the Coulombexchange within the Kohn-Sham approach [18] can im-prove the values for the dipole polarizabilities. It remainsan open question to which extent an improved treatmentof the Coulomb exchange will affect the positions of theunperturbed particle-hole excitations, and thus the de-tailed structures of the photoabsorption cross sectionsobtained in the present investigation.It can be concluded that direct decay of the giantdipole resonance in small neutral sodium clusters playsan important role in the fragmentation of the associatedstrength function, a process that seems to be essentiallyabsent in the case of the positively charged clusters. Theactual location and potential fragmentation of the dipolepeak in Na and Na is the result of a delicate balancebetween shell structure and residual interaction, whichmight be further unravelled by carrying photoabsorptionmeasurements at low temperatures. A detailed mappingof the strongly red shifted dipole peaks and of their struc-ture could provide some crucial tests of the exchange(plus correlation) part of the effective interaction betweenthe electrons in metal clusters.Discussions with W. de Heer, B.R. Mottelson and H.Nishioka are gratefully acknowledged. We wish to thankW.D. Knight and collaborators for providing us with themost recent experimental data prior to publication. Oneof us (C.Y.) wishes to acknowledge financial support dur-ing the course of this research from the INFN, sez. Mi-lano, from a NATO fellowship through the Greek Min-istry of National Economy, and from the Joint Institutefor Heavy Ion Research, where the calculations were com-pleted. ∗ Current address: Joint Institute for Heavy Ion Research,Oak Ridge National Laboratory, Oak Ridge, Tennessee37831. † Permanent address: Institut f¨ur Theoretische Physik,Universit¨at Regensburg, D-8400 Regensburg, West Ger-many.[1] W.A. de Heer et al, Phys. Rev. Lett. 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Leygnier,to be published[15] In any case, thermal fluctuations are expected to be lessimportant for charged clusters due to the fact that theelectrons are more tightly bound, and thus the associ- ated shell corrections[19] (Nilsson-Strutinsky model) areexpected to be larger leading to a stiffer potential energysurface in the space of cluster shapes, as compared to theneutral case.[16] K. Selby, M. Vollmer, J. Masui, V. Kresin, M. Kruger,W.A. de Heer, and W.D. Knight, Phys. Rev. B, to bepublished, and Contribution, Aix-en-Provence, Meetingon Clusters, July 1988[17] D.E. Beck, Phys. Rev. B (1984) 6935[18] P. Stampfli and K.H. Bennemann, Phys. Rev. A (1989) 1007[19] M. Brack et al, Rev. Mod. Phys.44